A Note on the Appearance of the Simplest Antilinear ODE in Several Physical Contexts
We review several one-dimensional problems such as those involving linear Schrödinger equation, variable-coefficient Helmholtz equation, Zakharov–Shabat system and Kubelka–Munk equations. We show that they all can be reduced to solving one simple antilinear ordinary differential equation <inline-...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-07-01
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| Series: | AppliedMath |
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| Online Access: | https://www.mdpi.com/2673-9909/2/3/24 |
| _version_ | 1797466035457097728 |
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| author | Dmitry Ponomarev |
| author_facet | Dmitry Ponomarev |
| author_sort | Dmitry Ponomarev |
| collection | DOAJ |
| description | We review several one-dimensional problems such as those involving linear Schrödinger equation, variable-coefficient Helmholtz equation, Zakharov–Shabat system and Kubelka–Munk equations. We show that they all can be reduced to solving one simple antilinear ordinary differential equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>u</mi><mo>′</mo></msup><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mover><mrow><mi>u</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow><mo>¯</mo></mover></mrow></semantics></math></inline-formula> or its nonhomogeneous version <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>u</mi><mo>′</mo></msup><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mover><mrow><mi>u</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow><mo>¯</mo></mover><mo>+</mo><mi>g</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><msub><mi>x</mi><mn>0</mn></msub></mfenced><mo>⊂</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>. We point out some of the advantages of the proposed reformulation and call for further investigation of the obtained ODE. |
| first_indexed | 2024-03-09T18:30:16Z |
| format | Article |
| id | doaj.art-5d6691a076a04ca4b0570079c0faa9ba |
| institution | Directory Open Access Journal |
| issn | 2673-9909 |
| language | English |
| last_indexed | 2024-03-09T18:30:16Z |
| publishDate | 2022-07-01 |
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| record_format | Article |
| series | AppliedMath |
| spelling | doaj.art-5d6691a076a04ca4b0570079c0faa9ba2023-11-24T07:33:07ZengMDPI AGAppliedMath2673-99092022-07-012343344510.3390/appliedmath2030024A Note on the Appearance of the Simplest Antilinear ODE in Several Physical ContextsDmitry Ponomarev0Institute of Analysis & Scientific Computing, Vienna University of Technology (TU Wien), Wiedner Hauptstrasse 8-10, 1040 Wien, AustriaWe review several one-dimensional problems such as those involving linear Schrödinger equation, variable-coefficient Helmholtz equation, Zakharov–Shabat system and Kubelka–Munk equations. We show that they all can be reduced to solving one simple antilinear ordinary differential equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>u</mi><mo>′</mo></msup><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mover><mrow><mi>u</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow><mo>¯</mo></mover></mrow></semantics></math></inline-formula> or its nonhomogeneous version <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>u</mi><mo>′</mo></msup><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mover><mrow><mi>u</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow><mo>¯</mo></mover><mo>+</mo><mi>g</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><msub><mi>x</mi><mn>0</mn></msub></mfenced><mo>⊂</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>. We point out some of the advantages of the proposed reformulation and call for further investigation of the obtained ODE.https://www.mdpi.com/2673-9909/2/3/24Helmholtz equationSchrödinger equationKubelka–Munk system |
| spellingShingle | Dmitry Ponomarev A Note on the Appearance of the Simplest Antilinear ODE in Several Physical Contexts AppliedMath Helmholtz equation Schrödinger equation Kubelka–Munk system |
| title | A Note on the Appearance of the Simplest Antilinear ODE in Several Physical Contexts |
| title_full | A Note on the Appearance of the Simplest Antilinear ODE in Several Physical Contexts |
| title_fullStr | A Note on the Appearance of the Simplest Antilinear ODE in Several Physical Contexts |
| title_full_unstemmed | A Note on the Appearance of the Simplest Antilinear ODE in Several Physical Contexts |
| title_short | A Note on the Appearance of the Simplest Antilinear ODE in Several Physical Contexts |
| title_sort | note on the appearance of the simplest antilinear ode in several physical contexts |
| topic | Helmholtz equation Schrödinger equation Kubelka–Munk system |
| url | https://www.mdpi.com/2673-9909/2/3/24 |
| work_keys_str_mv | AT dmitryponomarev anoteontheappearanceofthesimplestantilinearodeinseveralphysicalcontexts AT dmitryponomarev noteontheappearanceofthesimplestantilinearodeinseveralphysicalcontexts |